Hydrodynamic systems, mathematical modeling

In complex systems such as three-phase reactors, the methods of mathematical modeling cannot provide the required information for process design and scale-up since it is practically impossible to take into account all existing phenomena and safely predict the influence of hydrodynamics, heat and mass transfer, or kinetics on each other (Datsevich and Muhkortov, 2004). Thus, models are almost always approximate in nature. They are based on a number of assumptions that cannot be met during scale-up. So, it is not surprising that industrial unit designers do not completely trust the results obtained from mathematical modeling. Thus, several systems cannot be fully modeled mathematically and other methods for scale-up are followed. 

Among the many mathematical models of fluidized bed reactors found in the literature the model of Werther (J ) has the advantage that the scale-dependent influence of the bed hydrodynamics on the reaction behaviour is taken into account. This model has been tested with industrial type gas distributors by means of RTD-measurements (3)and conversion measurements (4), respectively. In the latter investigation (4) a simple heterogeneous catalytic reaction i.e. the catalytic decomposition of ozone has been used. In the present paper the same modelling approach is applied to complex reaction systems. The reaction system chosen as an example of a complex fluid bed reaction is the synthesis of maleic anhydride (Figure 1). 

This section will first deal with the phases in particle-fluid two-phase flow by developing a mathematical model to quantify local hydrodynamic states. This analysis will reveal the insufficiency of the conditions for the conservation of mass and momentum alone in determining the hydrodynamic states of heterogeneous particle-fluid systems, and calls for a methodology different from what is used in analyzing dilute uniform flow. For this purpose the concept of multi-scale interaction between particles and fluid and the principle of energy minimization are proposed. 

An experimental apparatus for continuously processing an aqueous stream containing an organic contaminant was designed and constructed. The criteria for choosing the contaminant/surfactant/ extraction solvent are discussed along with the system operating parameters such as the heptane/water ratio, the hydrodynamic conditions of the ultrafilter and the overall efficiency of the toluene separation. A mathematical model of the continuous system was also developed and evaluated. 

Reactive distillation occurs in multiphase fluid systems, with an important role of the interfacial transport phenomena. It is an inherently multicomponent process with much more complexity than similar binary processes. Multi-component thermodynamic and diffusional coupling in the phases and at the interface is accompanied by complex hydrodynamics and chemical reactions [4, 42, 43]. As a consequence, an adequate process description has to be based on specially developed mathematical models. However, sophisticated RD models are hardly applicable for plant design, model-based control and online process optimization. For such cases, a reasonable model reduction should be applied [44],... 

The mathematical model, which makes it possible to consider the influence of the hydrodynamic conditions of flow on the processes of mixing and chemical transformations of reacting substances in a liquid phase, assumes that the average flow characteristics of a multicomponent system can be described by the equations of continuum mechanics and will satisfy conservation laws. 

Currently used mathematical models of gas purification formed on simplified theoretical concepts of gas flow. They are not sufficiently taken into account the operational and design parameters of gas cleaning devices, as well as aero-hydrodynamic properties of gas-dispersed flows. These models cannot be used to search for the best options of integrated gas cleaning systems, as they show the properties of objects in a narrow range of parameters. We need more complete and appropriate mathematical models based on the study of the aerodynamics of gas and taking place in these events. 

The program ANSYS-14/CFX mathematical model of motion of polydispersed gas system. The character of the movement of dust particles under the influence of centrifugal force. This allowed choosing the desired hydrodynamic conditions and taking into account of the design under various conditions of scrubber. Experimental study of the hydrodynamic characteristics of the device to determine the empirical constants and test the adequacy of the hydrodynamic model. 

The hydrodynamics of the experimental system can be described theoretically. Such approach is very important for correct interpretation of the experimental results, and for their extrapolation for the conditions not attainable in the existing experimental system. With the mathematical model the parametric study of the system is also possible, what can reveal the most important factors responsible for the occurrence of the specific transport phenomena. The model was presented in details elsewhere [2]. It was based on the equations of the momentum and mass transfer in the simplified two-dimensional geometry of the air-water-surfactant system. Those basic equations were supplemented with the equation of state for the phopsholipid monolayer. The resultant set of equations with the appropriate initial and boundary conditions was solved numerically and led to temporal profiles of the surface density of the surfactant, T [mol m ], surface tension, a [N m ], and velocity of the interface. Vs [m s ]. The surface tension variation and velocity field obtained from the computations can be compared with the results of experiments conducted with the LFB. 

In order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

Driven nonlinear systems often tend to develop spatially periodic patterns. The underlying mathematical models usually permit a continuous set of linearly stable solutions. As a possible mechanism of selecting a specific pattern the principle of marginal stability is presented, being applicable to situations, where a propagating front leaves a periodic structure behind. We restrict our discussion to patterns on interfaces which are more easily accessible than three-dimensional structures, for example in hydrodynamic flow. As a concrete system a recently analyzed model for dendritic solidification is discussed. 

EF system EF consists of Water and Air subsystems. On the fluidity property basis, it is possible to describe these subs3 tems as various models of fluid. Models of fluid motion reflect following subsystems Wind, Waves and Current Mathematical models of interacting subsystems represented by equations of a rigid body motion in a fluid, equations of hydrodynamics and aerodynamics, equations of electric drives electrodynamics, equations of thruster s mechanics, equations, that describe processes in DP control systems. 

Consequently, modeling of a two-phase flow system is subject to both the constraints of the hydrodynamic equations and the constraint of minimizing N. Such modeling is a nonlinear optimization problem. Numerical solution on a computer of this mathematical system yields the eight parameters ... 

The major difficulty in predicting the viscosity of these systems is due to the interplay between hydrodynamics, the colloid pair interaction energy and the particle microstructure. Whilst predictions for atomic fluids exist for the contribution of the microstructural properties of the system to the rheology, they obviously will not take account of the role of the solvent medium in colloidal systems. Many of these models depend upon the notion that the applied shear field distorts the local microstructure. The mathematical consequence of this is that they rely on the rate of change of the pair distribution function with distance over longer length scales than is the case for the shear modulus. Thus... 

When Amundson taught the graduate course in mathematics for chemical engineering, he always insisted that all boundary conditions arise from nature. He meant, I think, that a lot of simplification and imagination goes into the model itself, but the boundary conditions have to mirror the links between the system and its environment very faithfully. Thus if we have no doubt that the feed does get into the reactor, then we must have a condition that ensures this in the model. We probably do not wish to model the hydrodynamics of the entrance region, but the inlet must be an inlet. One merit of the wave model we have looked at briefly is that both boundary conditions apply to the inlet. 

From the above summary of the application of hydrodynamic theories to amylose, it can be seen that discrepancies exist between the calculated and experimental parameters, expecially when the exponent in the Mark-Houwink equation is high, that is, a 1. Errors can arise in the experimental determinations of the various parameters and also in using the incorrect mathematical averages in the theoretical relations. (The hydro-dynamic theories are based on monodisperse polymer systems, a criterion which is rarely satisfied, and it is necessary to introduce somewhat arbitrary corrections for heterogeneity.) Furthermore, the model on which the theory is based, or the mathematical approximations introduced during the subsequent treatment, may be incorrect. 

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